TR-2004-19

A note on the degree prescribed factor problem

Abstract

The degree prescribed factor problem is to decide if a graph has a
subgraph satisfying given degree prescriptions at each vertex. Lovász, and
later Cornuéjols, gave structural descriptions on this problem in case the
prescriptions have no two consecutive gaps. We state the
Edmonds-Gallai-type structure theorem of Cornuéjols which is only implicit
in his paper. In these results the difficulty of checking the property of
criticality is near to the original problem. By extending a result of Loebl,
we prove that a degree prescription can be reduced to the edge and
factor-critical graph packing problem by a `gadget' if and only if all of
its gaps have the same parity. With this gadget technique it is possible to
obtain a description of the critical components. Finally, we prove two
matroidal results. First, the up hulls of the distance vectors of all
subgraphs form a contra-polymatroid. Second, we prove that the vertex sets
coverable by subgraphs F satisfying the degree prescriptions for all
v \in V(F) form a matroid, in case 1 is contained in all
prescriptions.